On the dependence of the reflection operator on boundary conditions for biharmonic functions

TL;DR

This paper explores how boundary conditions influence reflection formulas for biharmonic functions, generalizing the Schwarz symmetry principle and revealing different reflection structures depending on boundary conditions.

Contribution

It introduces new reflection formulas for biharmonic functions under various boundary conditions, extending classical symmetry principles to more complex boundary scenarios.

Findings

Reflection formulas vary with boundary conditions.

Point-to-point reflection occurs under certain conditions.

Continuous set reflection occurs under other boundary conditions.

Abstract

The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions $u(x,y)\in\mathbb{R}^2$ subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures. In particular, in the special case of the boundary, $\Gamma_0 :=\{y=0\}$, reflections are point to point when the given on $\Gamma_0$ conditions are $u=\partial_nu=0$, $u=\Delta u=0$ or $\partial_nu=\partial n\Delta u=0$, and point to a continuous set when $u=\partial_n\Delta u=0$ or $\partial_nu=\Delta u=0$ on $\Gamma_0$.